
where \(H_{{(s)},n}({\beta }_{(s)})\)  is the NLS objective function of the \(s^{\mbox{\tiny \rm th}}\)  model defined in (REF ), \(\Vert {\beta }_{(s)}\Vert _1=\sum _{i=1}^{p_s} |\beta _{i}|\)  is the penalty, and \(\lambda _s\)  is the model-specific tuning parameter.
The above optimization problem can be solved, e.g., by the coordinate descent algorithm [1]}.
Furthermore, we denote \(\widehat{{\mu }}^R_{(s)}={\bf K}_{(s)}({\widehat{\beta }^{R}}_{(s)}){\bf y}, \widehat{{\mu }}^R({\bf w})=\sum \nolimits _{s=1}^{{S_n}}w_s\widehat{{\mu }}^R_{(s)}\)  and \(L^R_n({\bf w})=\Vert \widehat{{\mu }}^R({\bf w})-{\mu }\Vert ^2\) .
